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 © | Dror Bar-Natan: Classes: 2015-16: Math 475 - Problem Solving Seminar: (26) Next: Blackboards for Tuesday March 1 Previous: Blackboards for Tuesday February 23

# Handout for March 1, "Exploit Symmetry"

Reading. Section 1.6 of Larson's textbook.

Next Quiz. Thursday March 3, on Larson's section 1.6, this handout, and more.

Makeup class for last Thursday will be on Friday March 4 at 12-1 at Bahen 1210. A video of that class will be available.

Problem 1 (Larson's 1.6.2).

1. Of all the rectangles which can be inscribed in a given circle, which has the greatest area?
2. Maximize $\sin\alpha+\sin\beta+\sin\gamma$, where $\alpha,\beta,\gamma$ are the angles of a triangle.
3. Of all the triangles of a fixed perimeter, which has the greatest area?
4. Of all the parallelepipeds of volume 1, which has the smallest surface area?
5. Of all the $n$-gons which can be inscribed in a given circle, which has the greatest area?
6. Dror adds: of all the $n$-gons with a given perimeter, which has the greatest area?

Problem 2 (Larson's 1.6.11). Prove that the product of four consecutive terms of an arithmetic progression, plus the fourth power of the common difference, is always a perfect square.

Problem 3. A function on the plane has the property that the sum of its values on the corners of any square (of any size and orientation) is zero. Prove that the function is the zero function.

Problem 4. Prove: You cannot colour the points of the plane with just three colours, so that no two points of distance 1 will be coloured with the same colour. What if you had four colours available?

Problem 5. Prove: A right-angled triangle with integer sides has area divisible by three.

Dror's Favourite "Exploit Symmetry" problem: Two players alternate placing $1\times 1$, $2\times 1$, and $4\times 1$ lego pieces on a circular table of diameter 100, with no overlaps. If a player cannot place a further piece, (s)he looses. Would you rather be the first to move of the second?

And as we're into circles, some circle problems:

1. Jackie is inside a circular arena, a lion can run on its perimeter, $V_L=4V_J$. Can Jackie escape?
2. Julian and a lion are inside a circular arena, $V_L=V_J$. Can Julian survive indefinitely?
3. Can you cover a disk of diameter 100 with 99 bands of size $100\times 1$? Overlaps are allowed, but no folding or cutting.
4. A spherical loaf of bread goes into a bread-slicing machine. Which slice will get the most crust?

Dror's Favourite "Explore Symmetry" topic: See http://drorbn.net/MUGS-1510: